📚 GCSE AQA Philosophy: Formula & Theorem Quick Reference Handbook | GCSE AQA 哲学：公式定理速查手册
Welcome to your go-to revision guide for GCSE AQA Philosophy. Here, we’ve distilled the essential arguments, ethical principles, and logical structures of the course into clear, memorable “formulae” and “theorems”. Each entry gives you the core reasoning in a precise, exam-ready format, paired with bilingual explanation to deepen understanding. Use this handbook to strengthen your recall and structure your essay answers with confidence.
欢迎使用这本 GCSE AQA 哲学速查手册。我们将课程中的核心论证、伦理原则和逻辑结构提炼成清晰易记的“公式”和“定理”。每一条都以精准、适合应试的格式呈现核心推理，并附有中英双语解释，以加深理解。借助本手册，你可以巩固记忆，自信地组织问答题答案。
1. The JTB Formula (Knowledge) | JTB 公式（知识定义）
Knowledge is traditionally analysed as Justified True Belief. The formula can be expressed as: S knows that p if and only if (i) p is true, (ii) S believes that p, and (iii) S is justified in believing that p.
知识传统上被分析为“确证的真信念”。公式可表示为：主体 S 知道命题 p，当且仅当 (i) p 为真，(ii) S 相信 p，且 (iii) S 对 p 的信念有充分理由。
K(S,p) ↔ (True(p) ∧ Bel(S,p) ∧ Justified(S,p))
The Gettier counterexamples challenge this by showing cases where JTB conditions are met but knowledge seems absent, leading to the ‘No False Lemmas’ addition: S’s justification must not depend on any false belief.
盖梯尔反例挑战了这一定义，表明在某些情况下，即便满足 JTB 条件，依然缺乏知识，由此产生了“无错误引理”的补充：S 的确证不得依赖于任何错误信念。
2. The Utilitarian Calculus | 功利主义计算（最大幸福原则）
Classical utilitarianism, formulated by Jeremy Bentham, holds that the morally right action is the one that produces the greatest happiness for the greatest number. The decision procedure can be expressed as a sum over all affected individuals.
边沁提出的古典功利主义认为，道德上正确的行为是为最大多数人带来最大幸福的行为。决策程序可表示为对所有受影响个体幸福量的总和。
Right(Action) ↔ Maximise Σ (Pleasureᵢ – Painᵢ) for all i
In Bentham’s felicific calculus, one weighs the intensity, duration, certainty, propinquity, fecundity, purity, and extent of pleasures and pains. Mill later refined this by distinguishing higher and lower pleasures.
在边沁的“幸福计算法”中，需权衡快乐与痛苦的强度、时长、确定性、远近、继生性、纯度及范围。密尔后来通过区分高级快乐与低级快乐完善了该理论。
3. The Categorical Imperative (Formula of Universal Law) | 绝对命令（普遍法则公式）
Kant’s first formulation of the categorical imperative commands: “Act only according to that maxim whereby you can at the same time will that it should become a universal law.” This provides a test for the moral permissibility of actions.
康德对绝对命令的第一表述是：“只依据那些你同时能够愿意其成为普遍法则的准则而行动。”这为检验行为的道德可允许性提供了一个标准。
Permissible(A) ↔ ∀x (Agent can coherently will that the maxim of A become a universal law)
If universalising the maxim leads to a contradiction in conception (e.g., making a false promise to secure a loan would undermine the whole practice of promising) or a contradiction in the will (you could not rationally will a world where no one helps others), the action is impermissible.
如果将该准则普遍化会导致概念上的矛盾（例如，为获得贷款而作虚假承诺会摧毁承诺制度本身）或意愿上的矛盾（你无法理性地意愿一个无人互助的世界），则该行为不被允许。
4. The Cosmological Argument Theorem (First Cause) | 宇宙论论证定理（第一因）
The Kalam version of the cosmological argument can be structured as a deductive argument: Everything that begins to exist has a cause. The universe began to exist. Therefore, the universe has a cause, which we identify as God.
卡拉姆版本的宇宙论论证可构建为演绎论证：一切开始存在的事物皆有原因；宇宙开始存在；因此，宇宙有一原因，我们称之为上帝。
P1: ∀x (Begins(x) → HasCause(x))
P2: Begins(Universe)
C: ∴ HasCause(Universe) (→ God)
Aquinas’s Third Way argues from contingency: if everything were contingent, at some time nothing would exist, and then nothing could come from nothing, so there must be a necessary being. Leibniz’s Principle of Sufficient Reason reformulates this necessity.
阿奎那第三路从偶然性出发：如果万物皆为偶然，那么在某些时刻可能无一物存在，而由无不能生有，故必然存在一个必然存在者。莱布尼茨的充足理由律重新表述了这种必然性。
5. The Design Argument Analogy (Paley’s Watch) | 设计论证类比（佩利的手表）
Paley’s teleological argument proceeds by analogy: Just as the complexity and purposefulness of a watch imply a watchmaker, the vastly greater complexity and purposefulness of the natural world imply a divine designer.
佩利的目的论论证通过类比进行：正如一只表的复杂性与目的性暗示着钟表匠，自然世界更为庞大的复杂性与目的性则暗示着一位神圣设计者。
(Watch : Watchmaker) analogous to (Universe : Divine Designer)
The argument is abductive: the existence of order, regularity, and apparent purpose in nature is best explained by the existence of an intelligent designer. Hume’s criticisms include the weakness of the analogy, the possibility of multiple designers, and the presence of imperfection in the world.
该论证是溯因推理：自然中的秩序、规律性与表面目的的最佳解释是存在一位智能设计者。休谟的批评包括类比力度不足、可能存在多位设计者以及世界存在不完善之处。
6. The Problem of Evil Logical Formula | 恶的问题的逻辑表达式
The logical problem of evil claims an inconsistency between the existence of an omnipotent, omnibenevolent God and the existence of evil. The classical inconsistent triad can be formalised as follows.
逻辑恶问题主张全能全善的上帝存在与恶的存在之间存在矛盾。经典的“不一致三元组”可形式化如下。
1. God is omnipotent (can prevent all evil)
2. God is omnibenevolent (wants to prevent all evil)
3. Evil exists
Classical theism cannot consistently hold all three.
The free will defence, associated with Alvin Plantinga, attempts to show that an omnipotent God might not be able to create a world containing moral good without the possibility of moral evil, and thus the three are logically compatible.
自由意志辩护（与普兰丁格相关）试图表明，全能上帝可能无法创造一个包含道德善却没有道德恶之可能性的世界，因此三者逻辑上可以相容。
7. Descartes’ Method of Doubt (Foundationalism) | 笛卡尔的怀疑方法（基础主义）
Descartes employs hyperbolic doubt to find an indubitable foundation for knowledge. He systematically doubts senses, the dream argument, and the evil demon, until he arrives at the Cogito: “I think, therefore I am” (Cogito, ergo sum).
笛卡尔采用极端怀疑方法，为知识寻找不可怀疑的基础。他系统性地怀疑感官、梦境论证和恶魔设定，直至抵达“我思故我在”的确定性。
If I doubt, I think → If I think, I must exist → Cogito, ergo sum
From this foundational certainty, he attempts to rebuild knowledge by proving God’s existence (via the trademark argument and ontological argument) and God’s non-deception, thus vindicating clear and distinct perceptions.
从这个基础确定性出发，他通过证明上帝存在（商标论证和本体论论证）及上帝并非骗子，来重建知识，从而确证清楚明白的知觉为真。
8. Hume’s Problem of Induction (The Inductive Fallacy) | 休谟的归纳问题（归纳谬误）
Hume argues that inductive reasoning – inferring the unobserved from the observed – cannot be rationally justified without circularity. The principle of the Uniformity of Nature (the future will resemble the past) stands in need of justification, but the only available justifications are themselves inductive.
休谟指出，归纳推理——由已观察推知未观察——无法在不陷入循环的情况下得到理性证成。自然齐一性原则（未来将与过去相仿）需要证成，但唯一可用的证成方式本身仍是归纳的。
Deduction: If premises true, conclusion necessarily true.
Induction: Premises support conclusion but do not guarantee it.
Hume: Any justification of induction relies on induction → circular.
Responses include Popper’s falsificationism: science does not proceed by induction but by conjectures and refutations; and pragmatic justifications that induction is simply the most successful method we have.
回应包括波普尔的证伪主义：科学并非依靠归纳前进，而是依靠猜想与反驳；以及实用主义证成：归纳仅仅是我们拥有的最成功方法。
9. Moore’s Naturalistic Fallacy Theorem | 摩尔的自然主义谬误定理
G.E. Moore argues in Principia Ethica that ‘good’ is a simple, unanalysable, non-natural property. Any attempt to define the good in naturalistic terms (e.g., pleasure, evolutionary fitness, what is desired) commits the naturalistic fallacy.
摩尔在《伦理学原理》中主张，“善”是一种单纯、不可分析、非自然的属性。任何试图以自然属性（如快乐、进化适应性、被欲求）来定义善的做法都犯了自然主义谬误。
For any natural property X: ‘Good’ ≠ X
The open question argument: “X is good, but is X really good?” remains meaningful.
This challenges ethical naturalism and utilitarianism. For Moore, moral reasoning must be based on intuition concerning what is intrinsically good (e.g., aesthetic enjoyments and friendships).
这挑战了伦理自然主义和功利主义。对摩尔而言，道德推理必须基于对内在善（如审美乐趣和友谊）的直觉。
10. Plato’s Theory of Forms (Universals Theorem) | 柏拉图的理型论（共相定理）
Plato posits that the sensible world is an imperfect copy of a transcendent realm of Forms (Ideas). True knowledge is not of particulars but of these eternal, unchanging universals. The Form of the Good is the ultimate principle illuminating all other Forms.
柏拉图主张，可感世界只是超越的理型（理念）世界的不完美摹本。真正的知识不是关于殊相，而是关于这些永恒、不变的共相。善的理型是照亮其他所有理型的最高法则。
Particular beautiful things participate in the Form of Beauty.
Knowledge (episteme) = grasp of Forms; mere opinion (doxa) = perception of particulars.
The simile of the cave illustrates the ascent from the shadows of ignorance to the sunlight of knowledge and the Form of the Good. Aristotle famously criticises the separation of Forms, arguing that forms are immanent in things.
洞穴比喻形象地呈现了从无知的阴影上升到知识阳光及善的理型的过程。亚里士多德则著名地批评了理型的分离，认为形式内在于事物之中。
11. The Mind-Body Problem Formulae | 心身问题公式
Dualism holds that mind and body are distinct substances (substance dualism) or distinct properties (property dualism). Physicalism identifies mental states with brain states. The ‘hard problem’ of consciousness is the challenge of explaining subjective, qualitative experience.
二元论认为心灵与身体是不同的实体（实体二元论）或不同的属性（属性二元论）。物理主义则将心理状态等同于大脑状态。意识的“困难问题”在于如何解释主观的、质的体验。
Substance Dualism: Mind ≠ Body (Descartes: res cogitans vs res extensa)
Identity Theory: Mental state M = Brain state B
Challenge: Qualia (the ‘what it is like’ of experience) seem irreducible.
Descartes’ divisibility argument: body is divisible, mind is indivisible; therefore, they are distinct. Criticisms include the causal interaction problem for dualism and multiple realisability for the identity theory.
笛卡尔的可分性论证：身体可分，心灵不可分，因此二者不同。批评包括二元论的因果交互问题，以及同一理论的多重可实现性反驳。
12. The Free Will Determinism Paradox | 自由意志与决定论悖论
If determinism is true (every event has a sufficient cause), then all actions are necessitated, apparently undermining moral responsibility. If indeterminism is true, random actions cannot ground responsibility either. This is the “free will problem”.
如果决定论为真（每一事件皆有充足原因），那么所有行为都必然发生，似乎摧毁了道德责任。如果非决定论为真，随机行为也无法构成责任的基础。这就是“自由意志问题”。
Hard Determinism: D → ¬Free Will → ¬Moral Responsibility
Libertarianism: ¬D + agent causation → Free Will
Compatibilism: Free Will = actions caused by one’s own desires, even if those desires are themselves determined.
Compatibilists (like Hume) redefine freedom as the liberty to act according to one’s will, uncoerced, and argue that determinism is actually a prerequisite for meaningful free action and responsibility.
相容论者（如休谟）将自由重新定义为按照自己意愿、不受强制地行动的能力，并主张决定论实际上是有意义的自由行为与责任的先决条件。
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